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Differentiation is mostly about correctly identifying which rule applies — once you spot the pattern, the actual calculation is usually quick. At its core, the derivative dy/dx tells you the instantaneous rate of change of y with respect to x, or geometrically, the slope of the tangent line to the curve at a given point. Here are 12 worked examples covering the four rules that cover almost every differentiation question at FSc/A-level: the power rule, product rule, quotient rule, and chain rule.

How to Decide Which Rule to Use

Before differentiating, it helps to scan the expression and ask a few quick questions:

• Is it just a single term raised to a power (like x⁵ or x⁻²)? → Use the power rule.
• Is it two separate functions multiplied together (like x²·sinx)? → Use the product rule.
• Is it one function divided by another (like sinx/x)? → Use the quotient rule.
• Is it a function "inside" another function (like sin(2x) or e^(x²))? → Use the chain rule.

Many harder problems actually combine two rules — for example, differentiating x²·sin(3x) needs both the product rule and the chain rule together. Recognising the outer structure first (product? quotient?) before worrying about the inner details keeps the working organised.

Power Rule (Examples 1–4)

1. Differentiate y = x⁵.
dy/dx = 5x⁴.

2. Differentiate y = 3x⁴ − 2x² + 7.
dy/dx = 12x³ − 4x.

3. Differentiate y = 1/x² (i.e. x⁻²).
dy/dx = −2x⁻³ = −2/x³.

4. Differentiate y = √x (i.e. x^(1/2)).
dy/dx = (1/2)x^(−1/2) = 1/(2√x).

Product Rule (Examples 5–7)

Rule: if y = u·v, then dy/dx = u'v + uv'.

5. Differentiate y = x²·sinx.
u=x², v=sinx. dy/dx = 2x·sinx + x²·cosx.

6. Differentiate y = x·eˣ.
u=x, v=eˣ. dy/dx = eˣ + x·eˣ = eˣ(1+x).

7. Differentiate y = (x+1)(x−3).
Expand first: y = x²−2x−3. dy/dx = 2x−2. (Product rule gives the same result: (1)(x−3)+(x+1)(1) = 2x−2.)

Quotient Rule (Examples 8–9)

Rule: if y = u/v, then dy/dx = (u'v − uv') / v².

8. Differentiate y = x/(x+1).
u=x, v=x+1. dy/dx = [(1)(x+1) − x(1)] / (x+1)² = 1/(x+1)².

9. Differentiate y = sinx/x.
dy/dx = (x·cosx − sinx) / x².

Chain Rule (Examples 10–12)

Rule: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x).

10. Differentiate y = (3x+1)⁴.
dy/dx = 4(3x+1)³ · 3 = 12(3x+1)³.

11. Differentiate y = sin(2x).
dy/dx = cos(2x) · 2 = 2cos(2x).

12. Differentiate y = e^(x²).
dy/dx = e^(x²) · 2x = 2x·e^(x²).

Common Mistakes Students Make

• Forgetting to multiply by the derivative of the inner function. In chain rule problems like sin(2x), students often write cos(2x) and stop, forgetting to multiply by the derivative of 2x (which is 2).
• Using the product rule when expanding would be simpler. For example 7, (x+1)(x−3), it's often faster to expand into a single polynomial and use the power rule than to apply the full product rule — both give the same answer, but expanding is less error-prone for simple binomials.
• Mixing up the quotient rule order. The numerator of the quotient rule is u'v − uv', not uv' − u'v. Swapping the order flips the sign of the entire answer.
• Not simplifying negative and fractional exponents correctly. Rewriting 1/x² as x⁻² before differentiating (rather than trying to differentiate a fraction directly) avoids a lot of avoidable errors.

Frequently Asked Questions

What does a derivative actually represent?
It represents the rate at which a quantity is changing at a specific instant — for a position function, the derivative is velocity; for a velocity function, the derivative is acceleration.

Do I need to memorise the product and quotient rules separately, or is one enough?
They're both worth memorising directly, since deriving the quotient rule from the product rule under exam time pressure wastes time. That said, understanding that the quotient rule is really the product rule applied to u·v⁻¹ can help you recall it if you blank.

Why do trigonometric and exponential derivatives always need the chain rule when there's an inner function?
Because functions like sin(x), cos(x), and eˣ are only "pure" when their input is simply x. As soon as the input becomes something like 2x or x², you're differentiating a composition of two functions, which is exactly what the chain rule is built to handle.

Practice More

These same rules extend directly to integration and optimization problems later in the syllabus — getting comfortable with them now pays off. Use the Formula Library on Calvo to keep the rules handy while practicing, and try re-deriving each rule from first principles at least once so the logic behind it sticks rather than just the formula.